# About finding roots of polynomials in specific domains

I have two polynomials $p(x)$ and $q(x)$ and I want to know if there are roots of the equation $\frac{p'}{p} = \frac{q'}{q}$ in the domain $(a,\infty)$ , where $a = max { roots(p), roots(q) }$

This is the same as asking for the roots of the polynomial, $p'q - pq' = 0$ in the same domain.

• Can something in Sage help?
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I would do as follows:

sage: r = p.derivative()*q - q*p.derivative()
sage: R_r = r.roots(RR)


This gives you the real roots of r in the variable R_r. Now to filter the roots that make sense to you, you need to compute the roots of p and q:

sage: R_p = p.roots(RR)
sage: R_q = q.roots(RR)


In R_p and R_q, you have the real roots of p and q, given as pairs with the root and the multiplicity. To find a:

sage: a = max([x[0] for x in R_p] + [x[0] for x in R_q])


And finally, the roots you want:

sage: R = [x in R_r if x[0] > a]

more

Thanks! Let me try this.

( 2015-06-12 19:56:35 +0100 )edit

If you just wrote "p.roots()" would it give all the roots? (and not just the reals)

( 2015-06-12 19:57:23 +0100 )edit

And what would be a command to check if R turned out to be empty? I need some kind of an error catching for that.

( 2015-06-12 19:58:16 +0100 )edit

If I am reading the code correctly and R is a list, then you could try len(R) == 0 . If I misunderstood, then nevermind.

( 2015-06-14 00:48:40 +0100 )edit

Alternatively

def are_there_roots(p, q):
return False

more

What if, for example, $p=q\neq 0$ or if $p'$ and $q'$ have a common root that is larger than $a$ ?

( 2015-06-12 14:56:14 +0100 )edit

I think it's hard for $p'$ and $q'$ to have a common root larger that the largest root of $p$ and $q$... I am impressed by how exact yet uninformative this answer is!

( 2015-06-12 16:15:03 +0100 )edit

@b-r-u-n-o wrote:

I think it's hard f=or p′ and q′ to have a common root larger that the largest root of p and q...

What about, say, $p=q=x^3+8$ whose single real root is -2, but whose derivative have root 0 ?

( 2015-06-12 17:41:18 +0100 )edit

max(roots(p), roots(q)) is ambiguous.

( 2015-06-12 22:55:34 +0100 )edit